Relative Velocity: Rain Man Problem

Understanding Relative Velocity

• Concept: Velocity of A with respect to B = Velocity of A - Velocity of B (relative to ground)
• Observer B sees the velocity of A minus the velocity of B

Rain Man Problem: Application in 2D

Example 1: Man Standing in Rain

• Rain speed: 40 km/hr downward
• Man’s speed: 0 (standing)
• Velocity of rain as seen by man = Velocity of rain - Velocity of man
  • Vr - Vm = 40j - 0 = 40j
• Man sees rain falling vertically downward
• Man should hold the umbrella vertically upward (opposite direction of rain)

Example 2: Man on Bicycle in Rain

• Rain speed: 40 km/hr downward
• Bicycle speed: 30 km/hr forward
• Calculate velocity of rain as seen by man on bicycle:
  • Velocity of rain: -40j
  • Velocity of man: 30i
  • Velocity of rain as seen by man: -40j - 30i
• Plot in x-y plane:
  • Minus 40j (downward) and minus 30i (backward)
  • Resultant velocity: Diagonally backward-downward
  • tan θ = 30/40 = 3/4 (right-angle triangle)
  • θ = 37 degrees (umbrella tilts forward by 37 degrees from vertical)

Example 3: Man on Trolley Moving Fast

• Rain speed: 60 km/hr downward
• Trolley speed: 80 km/hr forward
• Calculate velocity of rain as seen by man on trolley:
  • Velocity of rain: -60j
  • Velocity of man: 80i
  • Velocity of rain as seen by man: -60j - 80i
• Plot in x-y plane:
  • Minus 60j (downward) and minus 80i (backward)
  • tan θ = 80/60 = 4/3
  • θ = 53 degrees (umbrella tilts forward by 53 degrees from vertical)

Specific Scenarios

Scenario 1: Both Rain and Man Moving

• Rain speed: 50 km/hr at a 37-degree angle
• Bicycle speed: 40 km/hr forward
• Components of rain's velocity:
  • Horizontal (x): 50 cos(37)
  • Vertical (y): 50 sin(37)
  • Therefore, velocity of rain w.r.t ground: 40i - 30j
• Velocity of man: 40i
• Calculate velocity of rain as seen by man:
  • 40i - 30j - 40i = -30j
• Umbrella needs to be held vertically upward as man sees rain straight downward

Scenario 2: Woman on Inclined Plane

• Rain speed: 50 km/hr at a 53-degree angle
• Bike speed: 100 km/hr up the incline
  • Velocity components of rain: Horizontal component -50 cos 53, Vertical component in y direction -50 sin 53
  • Woman’s velocity components: Forward 80i, Upwards 60j
  • Calculate velocity of rain as seen by woman:
    • V_rain-ground - V_woman-ground = -30i - 40j - 80i + 60j = -110i - 100j
• Tan θ = 110/100
  • θ = tan inverse 11/10
  • Umbrella has to be adjusted accordingly

Summary and Key Points

• To determine appropriate umbrella angle, always calculate relative velocity
• Use vector components to simplify calculations
• Plot velocities on x-y plane for better visualization
• Consider angles and resultant velocities for accurate adjustment of umbrella